The
L(h,k)-Labelling Problem

In this page you can find a survey and annotated bibliography on the L(h,k)-labelling problem.

A continuously updated catalog of results on the
L(h,k)-labelling problem follows.
If you happen to notice some missing results, errors, not updated references, please write to
calamo AT di DOT uniroma1 DOT it.

I managed to prove that the problem of L(2,1)-edge labeling is NP-complete if \lambda≥5 and polynomial if \lambda≤4.

I'm going to present it on IWOCA (http://iwoca2015.di.univr.it/)
conference in Verona. After the conference it will be published as
post-proceedings.
I'll let you know when it will be published. Now we have a version on
arXiv: (http://arxiv.org/abs/1508.01014).

o
b>

09/23/2015

Dan Cranston

I'm writing to mention some of my papers that might be relevant. Most
of these results fit into the case of L(1,1)-labeling, although many
of them are actually for list-coloring or online list-coloring,
sometimes called paintability. (A few are so-called injective
coloring, which is L(0,1)-labeling.) You can download a preprint of
each paper either at http://www.people.vcu.edu/~dcranston/pubs/ or at
http://arxiv.org/a/cranston_d_1.html

Planar Graphs of Girth at least Five are Square (? + 2)-Choosable.
Marthe Bonamy, Daniel W. Cranston, and Luke Postle. (18pp) We prove a
conjecture of Dvorak et al. that there exists a constant C such that
for every planar graph with girth at least 5 and ? >= C the graph G^2
is (? + 2)-Choosable.

List-coloring Squares of Planar Graphs without 4-cycles and
5-cycles. Daniel W. Cranston and Bobby Jaeger. (15pp). Let G be a
planar graph with no 4-cycle and no 5-cycle (although 3-cycles are
allowed). We show that if ? >= 32, then G^2 is (? + 3)-Choosable.

List-coloring the Square of a Subcubic Graph. Daniel W. Cranston
and Seog-Jin Kim. Journal of Graph Theory. Vol. 57, January 2008,
pp. 65-87. We show that if G is a connected graph with ? = 3 and G is
not the Petersen graph, then G^2 is 8-choosable.

Painting Squares in ?^2-1 Shades. Daniel W. Cranston and Landon
Rabern. Submitted. (23pp) We generalize paper #3. We show that if G
is a connected graph with maximum degree ? and G is not a Moore graph,
then G is (?^2-1)-paintable (this is a stronger version of choosable).

About planar graphs, in section 4.6, page 18, you say Jonas proved in his P.h.D that
" L{1,1}(G) is at most 8 x Delta - 21 "
This seems wrong for me in the case Delta = 3, where we know L{1,1} is at least 6.
I couldn't obtain access for this thesis however.

In the end of the same paragraph, you relate this result :
" L{1,1}(G) is at most 5 x Delta "
And you precise this is better than the better asymptotic bound when Delta is at most 24.
However, you give reference before to a bound of 3 x Delta + 5, which is better than 5 x Delta in each case.

Also i published a paper you might fight interesting.
We give a bound for L{p,q} of graphs with bounded maximum average degree, which improve the known bounds on L{p,q} for planar graphs with girth 5, 6 and 7.

I have written a master thesis on the topic of complexity of L(h,k)-labelling. The thesis was advised my Jiri Fiala.
You might be interested in it.

x
b>

4/28/2011

Pawel Rzazewski

Concerning the exact algorithms for L(2,1)-labeling, mentioned in your survey, the result 3.23 was not published in the paper you refer to, but in the following:
K. Junosza-Szaniawski, P. Rz??ewski, On the Complexity of Exact Algorithm for L(2,1)-labeling of Graphs, to appear in Information Processing Letters, preliminary version in KAM Series 2010, 992

x

1/18/2010

J.-S. Sereni

In the journal version of our SODA 08 paper (currently under review), we adapted the proof
to obtain the same bounds for L(p,1)-labellings (and not only L(2,1)).
[A link to the preprint is:
http://iti.mff.cuni.cz/series/files/iti454.pdf
]
To sum-up, the results we obtain are the following [where our definition
of lambda(p,1) differs by 1 with the one of Griggs adn Yeh, which
explains why we have Delta^2+1 and not Delta^2].
Theorem: For any fixed integer p, there exists a constant Cp such that
for every integer Delta and every graph G of maximum degree Delta,
lambda(p,1)(G) <= Delta^2 + Cp.
This theorem is obtained as a corollary of the following.
Theorem: For any fixed integer p, there is a Delta_p such that
for every graph G of maximum degree Delta >= Delta_p,
lambda(p,1)(G) <= Delta^2 + 1.

x

5/11/2009

S. Noble

I thought you might be interested in a preprint of mine:
Eggemann, N., Havet, F., Noble, S.D. k-L(2,1)-Labelling for Planar
Graphs is NP-complete for k >= 4
available at
http://people.brunel.ac.uk/~mastsdn/L21NP-short.pdf
for your annotated bibliography.
As the title suggests we show that L(2,1)-labelling with span at most k
is NP-complete for planar graphs for every constant k>=4. This is best
possible and was only previously known for even values of k >= 8.

In your bibliography you mention that it is known for all values of k
but the paper you cite [62], doesn't show this. The relevant theorem
only explicitly deals with the case when k is part of the input,
although it clearly establishes the theorem for a constant k = 8. There
is a similarly titled conference paper by the same authors where they
give a different proof (for k part of the input) from which it might be
possible to establish the result for all large constant k, but the
precise position is far from clear.

x

May 2009

B. Sinaimeri

It appeared on SODA '08 a paper by Havet, Reed and Sereni proving the Griggs and Yeh's conjecture.

(speaking about paper: Jiri Fiala, Petr Golovach and Jan Kratochvil
"Computational complexity of the distance constrained labeling problem for trees"
accepted to ICALP '08) We've solved the problem we worked on last 5 years.

x
b>

10/16/2006

R.K. Yeh

a survey article of mine has been published ib Discrete Math. 306 (2006), 1217-1231

x
b>

07/18/2006

D. Sakai Troxell

you might want to update this reference (it used to be cited as a DIMACS Tech. Report):

This is just to announce that my joint papers with Teresa Jin
"Real number channel assignments for lattices" and
"Real number labelings for paths and cycles"
are now available on my homepage. They are now submitted.
They follow our paper
"Real number graph labelings with distance conditions"
which has been accepted by SIDMA. The recently revised, final version,
is now available on my homepage.
We have another, shorter, paper in a rough draft stage, which we plan
to complete and submit in the next few weeks. It will also be posted
on the Web when ready.

x
b>

5/27/2005

G. Agnarsson

Also, please be aware that there are some
pesky mistakes in our SODA'04 version:
For example, the outerplanar chordal graph
F_6=Hat(Hat(K_3)) of Delta=6 has chromatic number
7 and not 8 as claimed. There are other
mistakes as well...., We are sorry about this.

x
b>

5/6/2005

D. Troxell

(speaking about the L(2,1)-labeling on interval graphs)
(...)I only recall one paper years ago by Z.D. Shao
(Nanjing University) that touched on the subject of R-unit sphere graphs
and L(2,1)-labelings.

x
b>

2/4/2005

J. Fiala

(...) The graph used in the reduction for L(2,1)-coloring and in the radio
coloring paper is in fact a graph with exactly one prime graph in its
modular decomposition tree.
This is a P4 in the root of the tree. Hence the graph is obtained from a
P4 by substituting cographs into the four vertices of the P4.
The reasoning concerning LinMSOL does not work. Not even
Coloring is in LinMSOL. You need to fix the parameter to have
k-Coloring in LinMSOL.

x
b>

1/17/2005

X. T. Jin

you can download two of my joint papers with Dr. Griggs
in my homepage:
www.math.sc.edu/~jin2/myself.html
or in Dr. Griggs's homepage:
.

x
b>

11/10/20
04

G. Fertin

I am afraid there might be some intersection between sections 3.2 and 3.3. Indeed, any d-dimensional grid is a cartesian of d paths, and any d-dimensional torus is a cartesian product of cycles. Hence, separating the case grids/tori from the case cartesian product of paths/cycles might not be good.

- Section 2. L(1,1)-labeling: you can add also reference [AFNPS02]
where we provide a new local NP-completeness proof for this
problem on planar graphs.
- Section 2. L(2,1)-labeling: you can add also reference [TCS03,
not published yet, but accepted] where we provide a new local
NP-completeness proof for this problem on planar graphs.
- Section 3. Bounded treewidth graphs: I would like to mention
that the solution of [11, your citations] actually solves a
generalization of L(1,1)-labeling, the problem of coloring the
l-th power of a graph G, G^l.
- Section 3.6. Planar graphs: L(1,1)-labeling: Perhaps you could
mention that the result of [88, your citations], applies here
resulting to an 1,66 approximation algorithm.
- same section L(2,1)-labelling. The NP-completeness proof [42]
but the simpler one of [TCS03] prove the NP-completeness of the
problem for planar graphs of any degree, closing the open problem
of [11].
- I would like to contact you another research line for
L(1,1)-labelling done via the Probabilistic tools. In a work of
our group [FNPS03] we provide an upper bound for the problem of
for graphs of girth at most 7. A parallel work of [Alon02]
provides improved upper bounds for the problem for graphs of girth
g = 3; 4; 5 or 6,
and g \geq 7. In particular for the former case, they prove that
they bound the optimal number of colors needed to color the
square of a graph of girth g \geq 7 with \Theta(\Delta^2/lod
\Delta).
- Finally, I would like to contact you relevant research line, of
the L(h, k) problem on networks specified by succinct models,
including periodically or hiarchically specified networks, done by
our group, [AFNPS02, FNPS02,FNPS04].
You can also visit my home page, in
http://students.ceid.upatras.gr/~viki/
[TCS03] D.A. Fotakis, S.E. Nikoletseas, V.G. Papadopoulou and P.G.
Spirakis: NP-Completeness Results and Efficient Approximations for
Radiocoloring in Planar Graphs. Journal of Theoretical Computer
Science (to appear).
[AFNPS02] M. Andreou, D. Fotakis, S. Nikoletseas, V. Papadopoulou
and P. Spirakis: "On Radiocoloring Hierarchically Specified Planar
Graphs: PSPACE-completeness and Approximations". Mathematical
Foundations of Computer Science, August 2002 (MFCS 2002), LNCS
2420, pp. 81-92.
[FNPS02] D. Fotakis, S. Nikoletseas, V. Papadopoulou and P.
Spirakis: "Radiocolorings in Periodic Planar Graphs:
PSPACE-Completeness and Efficient Approximations for the Optimal
Range of Frequencies". Workshop on Graph Theoretic Concepts 2002
(WG 2002).
[FNPS04] Fotakis D. A., Nikoletseas S. E., Papadopoulou V. G.,
Spyrakis P. G.: "Radiocolorings in Periodic Planar Graphs:
PSPACE-Completeness and Efficient Approximations for the Optimal
Number of Frequencies". 1st International Conference "From
Scientific Computing to Computational Engineering", Mini-Symposium
Computational Mathematics & Applications. 8-10, Semp. 2004.
[FNPS03] S. Nikoletseas, V. Papadopoulou and P. Spirakis:
"Radiocoloring Graphs via the Probabilistic Method". 4th
Panhellenic Logic Symposium, 2003.
[Alon02] N. Alon and B. Mohar, The chromatic number of graph
powers, Combinatorics, Probability and Computing 11 (2002), 1-10.

x

10/26/2004

J. van
Leeuwen

the
revised version of our STACS paper appeared in the Computer Journal. The
revised version is the better reference now: H.L.Bodlaender, T. Kloks,
R.B. Tan, E.J. van Leeuwen. Approximations for Lambda-Colorings of Graphs.
The Computer Journals, 47 (2004), 193-204.

x

10/20/2004

M.
Salavatipour

the
journal version of our paper (that had appeared

in ESA'02) is
coming out soon. The title is"A
Bound on the Chromatic

Number of the Square of a Planar Graph". It has stronger results for

L(p,q)-labeling
for planar graphs than our conference version (almost by a

I
have done some survey work as well. The article has been send to Disc.
Math. about a year ago. Referees' report came to me on May (or maybe June),
but I did not do anything on that paper since I was full. However I have
revised the article and sent back to DM last month. You covered some
results that I did not and vice versa. Further several master students'(in Taiwan) theses are not included since I do not have a complete
list.

x

10/19/2004

J.
Kratochvil

If
a graph has bounded max degree then the L(2,1)

(and more
generally L(h,k)) span is also bounded, and hence the question

L(h,k)(G)\le
lambda can be expressed in MSOL and therefore is polynomial

for
graphs of bounded treewidth.

x

09/30/2004

J. Fiala

if
I am not mistaken - for fixed span l the L(p_1,...,p_k)labeling can

be expressed in
MSOL without quantifiers over the edge sets, so the

decision
problem whether L(p_1,...,p_k)(G) <= l can be decided in a

polynomial time
for graphs of bounded cliquewidth (by works of Coucelle